Theta提升和CM椭圆曲线

In this project, we plan to study the special values of the L-functions and symmetric square L-functions of CM elliptic curves, via the theta lifting method. By using the theta lifting from the 1-dimensional unitary group U(1) to U(1), we know that the value at 1 of the L-function of a CM elliptic curve is equal to the product of the norm of a theta function on the adele group of U(1) with a non-zero constant; also , by using the theta lifting from U(1) to the 2 dimensional quasi-split unitary group U(1,1), we know that the value at 2 of the symmetric square L-function of a CM elliptic curve is equal to the product of the norm of a theta function on the adele group of U(1,1) with a non-zero constant. We want to study the norms of these theta functions, thereby obtain information about the special values of the L-function and the symmetric square L-function of the elliptic curve.

在本项目中，我们计划用Theta提升的方法来研究CM椭圆曲线的L-函数和对称方L-函数的特殊值。 由从一维酉群U(1)到U(1)的Theta提升可知，CM椭圆曲线的L-函数在1处的值是U(1)的阿代尔群上某个Theta函数的范数与一个非零常数的乘积；而由从U(1)到二维拟分裂酉群U(1,1)的Theta提升可知CM椭圆曲线的对称方 L-函数在2处的值是U(1,1)的阿代尔群上某个Theta函数的范数与一个非零常数的乘积。我们想研究这些Theta函数的范数，从而得到关于椭圆曲线的L-函数和对称方L-函数的特殊值的信息。